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I'd like to see an extension of this discussion of the age-old chi-sq vs. Fisher's exact test debate, broadening the scope a bit. There are many many tests for interactions in a contingency table, enough to make my head spin. I'm hoping to get an explanation of what test I should use and when, and of course an explanation as to why one test should be preferred over another.

My current problem is the classic $n \times m$ case, but answers regarding higher dimensionality are welcome, as are tips for implementing the various solutions in R, at least, in cases where it is non-obvious how to proceed.

Below I've listed all the tests I'm aware of; I hope by exposing my errors they can be corrected.

  • $\chi^2$. The old standby. There are three major options here:

    • The correction built into R for 2x2 tables: "one half is subtracted from all $|O-E|$ differences." Should I always be doing this?
    • "$N-1$" $\chi^2$ Test, not sure how to do this in R.
    • Monte Carlo simulation. Is this always best? Why does R not give me df when I do this?
  • Fisher's exact test.

    • Traditionally advised when any cell is expected to be <4, but apparently some dispute this advice.
    • Is the (usually false) assumption that the marginals are fixed really the biggest problem with this test?
  • Barnard's exact test

    • Another exact test, except I've never heard of it.
  • Poisson regression

    • One thing that always confuses me about glms is exactly how to do this significance tests so help on that would be appreciated. Is it best to do nested model comparison? What about a Wald test for a particular predictor?
    • Should I really just always be doing Poisson regression? What's the practical difference between this and a $\chi^2$ test?
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This is a good question, but a big one. I don't think I can provide a complete answer, but I will throw out some food for thought.

First, under your top bullet point, the correction you are referring to is known as Yates' correction for continuity. The problem is that we calculate a discrete inferential statistic:
$$ \chi^2=\sum\frac{(O-E)^2}{E} $$
(It is discrete because, with only a finite number of instances represented in a contingency table, there are a finite number of possible realized values that this statistic can take on.) Notwithstanding this fact, it is compared to a continuous reference distribution (viz., the $\chi^2$ distribution with degrees of freedom $(r-1)(c-1)$). This necessarily leads to a mismatch on some level. With a particularly small data set, and if some cells have expected values less than 5, it is possible that the p-value could be too small. Yates' correction adjusts for this.

Ironically, the same underlying problem (discrete-continuous mismatch) can lead to p-values that are too high. Specifically, the p-value is conventionally defined as the probability of getting data that are as extreme or more than the observed data. With continuous data, it is understood that the probability of getting any exact value is vanishingly small, and thus we really have the probability of data that are more extreme. However, with discrete data there is a finite probability of getting data just like yours. Only calculating the probability of getting data more extreme than yours yields nominal p-values that are too low (leading to increased type I errors), but including the probability of getting data the same as yours leads to nominal p-values that are too high (which would lead to increased type II errors). These facts prompt the idea of the mid p-value. Under this approach, the p-value is the probability of data more extreme than yours plus half the probability of data just the same as yours.

As you point out, there are many possibilities for testing contingency table data. The most comprehensive treatment of the pros and cons of the various approaches is here. That paper is specific to 2x2 tables, but you can still learn a lot about the options for contingency table data by reading it.

I also do think it's worth considering models seriously. Older tests like chi-squared are quick, easy, and understood by many people, but do not leave you with as comprehensive an understanding of your data as you get from building an appropriate model. If it is reasonable to think of the rows [columns] of your contingency table as a response variable, and the columns [rows] as an explanatory / predictor variables, a modeling approach follows quite readily. For instance, if you had just two rows, you can build a logistic regression model; if there are several columns, you could use reference cell coding (dummy coding) to build an ANOVA-type model. On the other hand, if you have more than two rows, multinomial logistic regression can be used in the same manner. Should your rows have an intrinsic order, ordinal logistic regression would yield superior performance to multinomial. The log-linear model (Poisson regression) is probably less relevant unless you have contingency tables with more than two dimensions, in my opinion.

For a comprehensive treatment of topics like these, the best sources are the books by Agresti: either his full-scale treatment (more rigorous), his intro book (easier but still comprehensive and very good), or possibly also his ordinal book.

Update: Just for the sake of the completeness of list of possible tests, it occurs to me that we can add the likelihood ratio test (often called the '$G^2\text{-test}$'). It is:
$$ G^2=\sum O\cdot\text{ln}\left(\frac{O}{E}\right) $$
This is also distributed as a chi-squared, and will almost always yield the same decision. The realized values of the two statistics will typically be similar, but slightly different. The question of which will be more powerful in a given situation is quite subtle. I gather it is the default choice by tradition in some fields. I do not necessarily advocate it's use over the traditional test; I'm only listing it for completeness, as I say.

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    $\begingroup$ That was a great explanation of the underlying problem, thanks! Also I've been told in the past that Agresti's text is a great resource so I'll check it out. $\endgroup$ – JVMcDonnell Jun 1 '12 at 16:32
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I will try to address some of your questions as best as I can from my perspective. First the Fisher-Irwin Test is just another name for Fisher's exact test. Except for the fact that it is sometimes computationally intense I generally prefer to use the Fisher test. If there is any issue with this test it is conditioning on the marginal totals. The beauty of the test is that under the null hypothesis the set of contingency tables with the same marginal totals as the observed table has a hypergeometric distribution. Some people argue that they don't see the rationale for restricting consideration to tables with the same marginal totals.

Pearson's chi-square test is very commonly used to test for association in contingency tables. Like many other tests it is approximate and so the significance level is not always accurate. Cochran showed that in small samples when some cells are very sparse (e.g. containing fewer than 5 cases in some cells) the approximation will be poor.

There are many other approximate tests. Typically when applying Fisher's test using SAS I get the results from all of these tests and they usually give almost the same results. But Fisher's test is always exact conditional on the marginal totals.

Regarding Poisson regression, that is a model that relates the categorical variables to the cell totals. Like any model it depends on a set of assumptions. Most important is that the cell counts follow a Poisson distribution which means that the mean number of counts is equal to its variance. This is not generally true for cell count distributions. In the case of overdispersion (variance greater than mean) a negative binomial model might be more appropriate.

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  • $\begingroup$ "Fisher-Irwin Test is just another name for Fisher's exact test" ... aha, this makes this comment less confusing to me, thanks! $\endgroup$ – JVMcDonnell May 29 '12 at 17:08
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    $\begingroup$ Your answer hasn't really reduced my confusion about when to do these things. I guess one of the things I was hoping to hear about are to what extent the issues with chi^2 are solvable by monte carlo simulation or corrections, etc; or the extent to which it can be superseded by glms. So I'm just going to leave this open for a bit to see if I can get more bites. But if nobody weighs in after a bit I'll accept your answer. $\endgroup$ – JVMcDonnell May 31 '12 at 17:22
  • $\begingroup$ For Fisher and Chi-square I think I told you when you can use chi square. If you accept Fisher's idea that you should always condition on the marginal totals, Fisher's test is always applicable. But if you don't accept that then I guess you would have to pick an unconditional test. As for the other battery of tests available i don't know anything about their properties and hence can't really advise you when to use them. Form experience I have seen cases where it mattered because the result are usually in close agreement. $\endgroup$ – Michael R. Chernick May 31 '12 at 17:33
  • $\begingroup$ Is it really true that Fisher thought that "you should always condition on the marginal totals"? This assumption is only valid when the marginal totals are fixed. In the lady tasting tea example, the lady knows that 5 are milk-first and 5 are milk-last. But it's more common in experiments that there is no force enforcing the marginals. Consider the case of flipping two coins 10 times each. When 5 heads roll around the coin doesn't start giving tails to preserve the marginals. In such cases it's been documented that Fisher's is highly conservative. That's why I'm interested in the alternatives. $\endgroup$ – JVMcDonnell May 31 '12 at 17:46
  • $\begingroup$ Yes. It is my understanding that Fisher believed in choosing refernce distributions that use information from the given data. So he would think that no matter how the mariginal totals came about your observed data should only be compared to data that would have happened under the null hypothesis that followed the contstraints on the data namely the given marginal totals. As with other ideas that Fisher had this was controversial. $\endgroup$ – Michael R. Chernick May 31 '12 at 18:19

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